Domain was migrated from the one Plesk to another. SmarterMail Web Client on destination server is used.
Upon login to SmarterMail Web Client on destination server and it is shown that all dates of messages are set equal to the migration data/time on server.
Such behavior (incorrect mail timestamps after migration) is considered as software issue, the internal id is PPPM-1189.
This issue was fixed in MU#45 for Plesk 11.5: Plesk 11.5.30 MU#45
Apply the latest Plesk micro-updates.